Energy, Intensity and Momentum of Electromagnetic Waves: Quick Revision

A brief note on Energy, Intensity and Momentum of Electromagnetic Waves. Read it before a test. Try each question at the end before you open its answer.

Energy and intensity

• Energy density: $u_E = u_B$. $\bar{u} = \tfrac{1}{2}\varepsilon_0 E_0^2 = B_0^2/2\mu_0$.
• Intensity: $I = \bar{u}c = \tfrac{1}{2}\varepsilon_0 cE_0^2 = \varepsilon_0 cE_{rms}^2$.
• Fields from a power: $I = P/4\pi r^2$ (point source) or $P/\pi a^2$ (beam). $E_0 = \sqrt{2I/\varepsilon_0 c}$, $B_0 = E_0/c$.
• Poynting vector: $\vec{S} = \vec{E}\times\vec{B}/\mu_0$. Along the direction of travel. Average $= I$.
• Falls with distance: $I \propto 1/r^2$, $E_0 \propto 1/r$.

Momentum and pressure

• Momentum: $p = U/c$.
• Absorbed: $F = IA/c$, pressure $I/c$.
• Reflected straight back: $F = 2IA/c$, pressure $2I/c$. Partly: $(1 + \rho)I/c$.
• Oblique mirror, beam of power $P$: $F = 2P\cos\theta/c$, along the normal.
• Sunlight: $4.54\ \mu\text{Pa}$ (black), $9.07\ \mu\text{Pa}$ (mirror).

Usual calculations

• Pulse: $P = U/\tau$, $F = P/c$.
• Energy in a length $L$ of a beam: $U = PL/c$.
• Displacement current: $I_d = \varepsilon_0 A\,dE/dt$.

A worked example

Sunlight of intensity $6.0$ W/m² falls straight on a mirror in a laboratory test. Find the peak magnetic field of the light and the pressure on the mirror.

1. Electric field. $E_0 = \sqrt{\dfrac{2 \times 6.0}{2.655 \times 10^{-3}}}\ \text{V/m} = 67.2$ V/m.
2. Magnetic field. $B_0 = \dfrac{E_0}{c} = 224.1$ nT.
3. Pressure. $p = \dfrac{2I}{c} = \dfrac{2 \times 6.0\ \text{W/m}^2}{3.00 \times 10^{8}\ \text{m/s}} = 40.0$ nPa.

Practise with new numbers

Check yourself

1. In an electromagnetic wave, what is the ratio of the electric energy density to the magnetic energy density?

2. Why does a mirror feel twice the force of a black surface in the same light?

3. If the distance from a point source is doubled, how do the intensity and the peak electric field change?

Sources

• OpenStax, University Physics Volume 2, Chapter 16 (Electromagnetic Waves).
• NCERT Physics Class 12, Chapter 8 (Electromagnetic Waves).